Theorem exmidontri2or 7310

Description: Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)

Assertion

Ref	Expression
exmidontri2or	|- (EXMID <-> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )

Distinct variable group:   x, y

Proof of Theorem exmidontri2or

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exmidontriim 7292	. . 3 |- (EXMID -> A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) )
2		onelss 4422	. . . . . . . 8 |- ( y e. On -> ( x e. y -> x C_ y ) )
3	2	adantl 277	. . . . . . 7 |- ( ( x e. On /\ y e. On ) -> ( x e. y -> x C_ y ) )
4		orc 713	. . . . . . 7 |- ( x C_ y -> ( x C_ y \/ y C_ x ) )
5	3, 4	syl6 33	. . . . . 6 |- ( ( x e. On /\ y e. On ) -> ( x e. y -> ( x C_ y \/ y C_ x ) ) )
6		eqimss 3237	. . . . . . . 8 |- ( x = y -> x C_ y )
7	6, 4	syl 14	. . . . . . 7 |- ( x = y -> ( x C_ y \/ y C_ x ) )
8	7	a1i 9	. . . . . 6 |- ( ( x e. On /\ y e. On ) -> ( x = y -> ( x C_ y \/ y C_ x ) ) )
9		onelss 4422	. . . . . . . 8 |- ( x e. On -> ( y e. x -> y C_ x ) )
10	9	adantr 276	. . . . . . 7 |- ( ( x e. On /\ y e. On ) -> ( y e. x -> y C_ x ) )
11		olc 712	. . . . . . 7 |- ( y C_ x -> ( x C_ y \/ y C_ x ) )
12	10, 11	syl6 33	. . . . . 6 |- ( ( x e. On /\ y e. On ) -> ( y e. x -> ( x C_ y \/ y C_ x ) ) )
13	5, 8, 12	3jaod 1315	. . . . 5 |- ( ( x e. On /\ y e. On ) -> ( ( x e. y \/ x = y \/ y e. x ) -> ( x C_ y \/ y C_ x ) ) )
14	13	ralimdva 2564	. . . 4 |- ( x e. On -> ( A. y e. On ( x e. y \/ x = y \/ y e. x ) -> A. y e. On ( x C_ y \/ y C_ x ) ) )
15	14	ralimia 2558	. . 3 |- ( A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) -> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )
16	1, 15	syl 14	. 2 |- (EXMID -> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )
17		ontri2orexmidim 4608	. . . 4 |- ( A. x e. On A. y e. On ( x C_ y \/ y C_ x ) -> DECID z = { (/) } )
18	17	adantr 276	. . 3 |- ( ( A. x e. On A. y e. On ( x C_ y \/ y C_ x ) /\ z C_ { (/) } ) -> DECID z = { (/) } )
19	18	exmid1dc 4233	. 2 |- ( A. x e. On A. y e. On ( x C_ y \/ y C_ x ) -> EXMID )
20	16, 19	impbii 126	1 |- (EXMID <-> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   \/ w3o 979   = wceq 1364   e. wcel 2167   A.wral 2475   C_ wss 3157   (/)c0 3450   {csn 3622  EXMIDwem 4227   Oncon0 4398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-uni 3840  df-tr 4132  df-exmid 4228  df-iord 4401  df-on 4403  df-suc 4406

This theorem is referenced by:  onntri52  7311